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A simple method is proposed, for incremental static analysis of a set of inter-colliding particles, simulating 2D flow. Within each step of proposed algorithm, the particles perform small displacements, proportional to the out-of-balance forces, acting on them. Numerical experiments show that if the liquid is confined within boundaries of a set of inter-communicating vessels, then the proposed method converges to a final equilibrium state. This incremental static analysis approximates dynamic behavior with strong damping and can provide information, as a first approximation to 2D movement of a liquid. In the initial arrangement of particles, a rhombic element is proposed, which assures satisfactory incompressibility of the fluid. Based on the proposed algorithm, a simple and short computer program (a “pocket” program) has been developed, with only about 120 Fortran instructions. This program is first applied to an amount of liquid, contained in a single vessel. A coarse and refined discretization is tried. In final equilibrium state of liquid, the distribution on hydro-static pressure on vessel boundaries, obtained by proposed computational model, is found in satisfactory approximation with corresponding theoretical data. Then, an opening is formed, at the bottom of a vertical boundary of initial vessel, and the liquid is allowed to flow gradually to an adjacent vessel. Almost whole amount of liquid is transferred, from first to second vessel, except of few drops-particles, which remain, in equilibrium, at the bottom of initial vessel. In the final equilibrium state of liquid, in the second vessel, the free surface level of the liquid confirms that the proposed rhombing element assures a satisfactory incompressibility of the fluid.

In Computational Fluid Mechanics, recently, a combination of grid method (in Eulerian co-ordinates) with a mesh-free particle method (in Lagrangian co-ordinates) is recommended [

Recently, the movement of a fluid is treated by incremental static analysis [

Current work has two motivations: 1) To develop a simple algorithm for incremental static analysis of inter- colliding particles, which approximates dynamic behavior with strong damping and is simpler than particle dynamics, which creates artificial high frequency oscillations which have to be suppressed by an additional technique. We can see SPH [

In 2D (two dimensional space), a single vessel (

Instead of dynamic analysis of particles which creates artificial high frequency oscillations, a simple incremental static analysis is preferred here, which approximates dynamic behavior with strong damping.

Within each step of incremental static analysis, every particle performs small displacements

The algorithm, based on proposed incremental static analysis, for the movement of a set of inter-colliding point-particles, simulating 2D flow, is briefly shown by the flow-chart of

The boundary conditions are given, that is, position, configuration and dimensions of linear boundaries of single vessel (

The initial conditions of the particles (x, y co-ordinates in 2D) are given, as shown in

For all out-of-balance nodal forces

Then, every particle performs small displacements

From the new positions (x, y) of the particles, the boundary-particle distances

The out-of balance force, acting on every particle, is found, as shown in

where

At the end of present step of algorithm, the following can be received, as output data: For every particle, displacements

cent to boundaries, with

If maximum absolute nodal force

Based on the above-described algorithm, for incremental static analysis of a set of inter-colliding point-particles, simulating 2D flow, a simple and short computer program (“pocket” program) has been developed, with totally only about 120 Fortran instructions (50 for main program of incremental static analysis + 50 for subroutine of boundary-particle

The above program runs with the version Force 2.0 of Fortran, whose Compiler is free available in Google, even in Internet Cafés.

The particles are initially arranged, in such a way, so that to form elementary rhombs, with particles at their four external nodes, as shown in

where

For the limit deformation

that is, the area of elementary rhomb has been reduced by only 8.6%, as shown in

After this limit deformation, that is, for

So, the proposed rhombic element, in the initial arrangement of particles, exhibits a satisfactory incompres- sibility, as will also be confirmed in the following applications.

Input data. In 2D (two dimensional space), a square vessel, with sides 100 cm, is considered, as shown in

arranged, in such a way, so that to form elementary rhombs, which have an in-compressibility property, as described in previous Section 5. There are 10 rows of 9 particles, alternated with 9 rows of 10 particles, that is totally 2 × 9 × 10 = 180 particles. The weight of every particle is

In

The boundary-particle

The parameter

On the other hand, the

The step-length of algorithm,

The particles are initially sparsely arranged, so that all

Output data. The present first application run by the previously described, in Sections 3, 4, simple and short computer program, for incremental static analysis of a set of inter-colliding point-particles, simulating 2D flow. And, in about 2.000 steps of algorithm and only 5.0 sec of computing time, the amount of liquid, simulated by 180 particles (coarse mesh) reached to final equilibrium state, shown in

of particles, where all out-of-balance nodal forces, in absolute values

It is observed, in

the external repulsive reactions at boundaries, obtained by coarse discretization, are compared to corresponding forces, obtained from theoretical hydro-static pressure distribution, shown in

Input data. A more refined mesh is tried, for the same previous problem of first application (same amount of liquid, in same 2D vessel). The initial positions of particles are shown in

ticles, alternated with 18 rows of 19 particles, that is totally 2 × 18 × 19 = 684 particles. Weight of a particle is

Output data. The present second application, with single vessel and refined mesh, has also run by the same simple and short computer program, and, in about 8000 steps of algorithm and 1.5 min of computing time, reached to final equilibrium state, where all out-of-balance nodal forces are, in absolute values

A slightly improved approximation, between computational and theoretical data is achieved by refined mesh, as shown in

After the obtained final equilibrium state of liquid, in the initial single vessel, of first application, now, an opening, with height 20 cm, is formed at the bottom of right vertical boundary, and the liquid is allowed to flow gradually, by incremental static analysis, to an adjacent vessel, communicating with the first one, as shown in

Application | Particles number | Steplength Δu (mm) | Steps number | Computing Time |
---|---|---|---|---|

1. Single vessel coarse mesh | 180 | 0.1 | 2,000 | 5.0sec |

2. Single vessel refined mesh | 684 | 0.025 | 8,000 | 1.5min |

3. Incomplete flow coarse mesh | 180 | 0.1 | 100,000 | 5.0min |

4. Complete flow coarse mesh | 180 | 0.1 | 300,000 | 15min |

5. Incomplete flow refined mesh | 684 | 0.025 | 400,000 | 1.5hr |

6. Complete flow refined mesh | 684 | 0.025 | 1,200,000 | 4.5hr |

However, there is not yet an equilibrium state, as significant out-of-balance nodal forces still remain, much larger, in absolute values

The external repulsive reactions, at the boundaries, are also noted in

In previous third application, the flow, from initial to adjacent vessel, was in-complete and there was not an equilibrium state. Now, the flow, from first to second vessel, continues. In about 300,000 steps of algorithm and 15 min of computing time, from beginning of first application, all out-of-balance nodal forces are, in absolute values

almost whole amount of liquid has been transferred to second vessel, except of few, fourteen, drops-particles, which remain, in equilibrium, at the bottom of first vessel, as shown in

The above state is obviously a few steps of algorithm before final equilibrium, in second vessel. However, it is preferred to demonstrate, here, this state, because final equilibrium, in a vessel, has already been demonstrated in two first applications.

The initial vessel, with width 100 cm, is narrower than the adjacent one, which has double width, 200 cm. On the other hand, in final equilibrium state of first vessel (

If the present fourth application had run with the refined mesh, the input-output would be complicated, with 684 particles and step-length Δu = 0.025 mm, compared to 180 particles and Δu = 0.1 mm of coarse mesh used. And a lot of algorithm steps, 1,200,000, and 4.5 hr of computing time would be required, by refined mesh, much more than corresponding values of 300,000 steps and only 15 min of computing time, of the coarse mesh used.

In

A simple incremental static algorithm is proposed, for 2D flow, simulated by inter-colliding point-particles (with zero area). Within each step of proposed algorithm, every particle performs a small displacement, proportional to the out-of-balance force, acting on it. Numerical experiments show that, if the liquid is confined within boundaries of a set of inter-communicating vessels, the algorithm converges to a final equilibrium state.

The proposed incremental static method approximates dynamic behavior of a liquid with strong damping. So, artificial high frequency oscillations are suppressed. And there is no more need for an additional technique to suppress them (see SPH_Smoothed Particle Hydrodynamics).

A rhombic element, in the initial arrangement of particles, is proposed, with particles at its four external nodes which assure sufficient incompressibility of the liquid.

Based on the proposed simple incremental static algorithm, for a set of inter-colliding point-particles, simulating 2D flow, a simple and short computer program has been developed, with totally only about 120 Fortran instructions.

The above simple and short computer program is first applied to an amount of liquid, contained in a singe vessel. A coarse discretization and a refined one are tried. In final equilibrium state, the hydrostatic pressure distribution, on vessel boundaries, obtained by proposed computational model, is compared to corresponding theoretical data, and is found in satisfactory approximation with them.

At the bottom of a vertical boundary of initial vessel, an opening is formed, and the liquid is allowed to flow gradually, by incremental static analysis, to an adjacent vessel. Almost whole amount of liquid is transferred to second vessel, except of few drops-particles, which remain, in equilibrium, at the bottom of first vessel; this is reasonable and is expected. In the final equilibrium state, in second vessel, when all out-of-balance forces, acting on particles, are in absolute values, less than 0.1 N, the free surface level of liquid, confirms that a satisfactory fluid incompressibility is assured by the proposed rhombic element.

If the large application, of complete flow from initial to adjacent vessel, had run by the refined discretization, input-output would be complicated and excessive computing time would be consumed.

Panagis G. Papadopoulos,Christopher G. Koutitas,Panos P. Lazaridis, (2016) Incremental Static Analysis of 2D Flow by Inter-Colliding Point-Particles and Use of Incompressible Rhombic Element. Open Journal of Civil Engineering,06,397-409. doi: 10.4236/ojce.2016.63034